An augmented second
is a major second
raised a half step
(or a semitone
, or half a tone
- all are synonymous
). Thus, if two notes are an augmented second apart, there are three semitones (one an a half tones) separating them.
An augmented second is enharmonic to a minor third, i.e. it sounds the same, but has a different name.
So what is the difference between an augmented second and a minor third?
The difference is a terminological one - a second (in western music) is defined as the distance between two noted in a scale. Two adjacent notes in a scale generally have different names, disregarding sharps or flats. So in A major, you have A, B, C#, D, E, F#, and G#. There are no repeats either.* You would not write A, B, Db, D, E, F#, G#, as this is musically wrong. You can only have one of each note. If you play the notes, though, it sounds like (is?) an A major scale.
So, between adjacent notes we have seconds. Between A and B we have a major second, between C# and D a minor second, and so forth. But what if we have C and D#, for example? There are three semitones, but they are not a third apart - they are only a 'step' apart - a second. So it's called an augmented second.
This is true of stand alone notes (i.e. out of context), as well as in the context of a scale. In correct music theory, between a B and a D we have a minor third, but between Cb and D we have an augmented second.
The best example of when this 'really' occurs is in the harmonic minor scale. Between the sixth and seventh degrees, we have an augmented second. In A harmonic minor, we have A, B, C, D, E, F and G#. Between F and G# there exists an augmented second.
*This is true for church modes, not for "unnatural" scales, such as the symmetrical diminished scale, which has 8 notes, so there has to be at least one repeat.